Theorem ipasslem1 26160

Description: Lemma for ipassi 26170. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ip1i.1	|- X = ( BaseSet ` U )
ip1i.2	|- G = ( +v ` U )
ip1i.4	|- S = ( .sOLD ` U )
ip1i.7	|- P = ( .iOLD ` U )
ip1i.9	|- U e. CPreHil OLD
ipasslem1.b	|- B e. X

Assertion

Ref	Expression
ipasslem1	|- ( ( N e. NN0 /\ A e. X ) -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) )

Proof of Theorem ipasslem1

Dummy variables j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0cn 10846	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
2		ax-1cn 9580	. . . . . . . . . . . 12 |- 1 e. CC
3		ip1i.9	. . . . . . . . . . . . . 14 |- U e. CPreHil OLD
4	3	phnvi 26145	. . . . . . . . . . . . 13 |- U e. NrmCVec
5		ip1i.1	. . . . . . . . . . . . . 14 |- X = ( BaseSet ` U )
6		ip1i.2	. . . . . . . . . . . . . 14 |- G = ( +v ` U )
7		ip1i.4	. . . . . . . . . . . . . 14 |- S = ( .sOLD ` U )
8	5, 6, 7	nvdir 25940	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ ( k e. CC /\ 1 e. CC /\ A e. X ) ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
9	4, 8	mpan 668	. . . . . . . . . . . 12 |- ( ( k e. CC /\ 1 e. CC /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
10	2, 9	mp3an2 1314	. . . . . . . . . . 11 |- ( ( k e. CC /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
11	1, 10	sylan 469	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
12	5, 7	nvsid 25936	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ A e. X ) -> ( 1 S A ) = A )
13	4, 12	mpan 668	. . . . . . . . . . . 12 |- ( A e. X -> ( 1 S A ) = A )
14	13	adantl 464	. . . . . . . . . . 11 |- ( ( k e. NN0 /\ A e. X ) -> ( 1 S A ) = A )
15	14	oveq2d 6294	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k S A ) G ( 1 S A ) ) = ( ( k S A ) G A ) )
16	11, 15	eqtrd 2443	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G A ) )
17	16	oveq1d 6293	. . . . . . . 8 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( ( k S A ) G A ) P B ) )
18		ipasslem1.b	. . . . . . . . . . . . 13 |- B e. X
19		ip1i.7	. . . . . . . . . . . . . 14 |- P = ( .iOLD ` U )
20	5, 19	dipcl 26039	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) e. CC )
21	4, 18, 20	mp3an13 1317	. . . . . . . . . . . 12 |- ( A e. X -> ( A P B ) e. CC )
22	21	mulid2d 9644	. . . . . . . . . . 11 |- ( A e. X -> ( 1 x. ( A P B ) ) = ( A P B ) )
23	22	adantl 464	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( 1 x. ( A P B ) ) = ( A P B ) )
24	23	oveq2d 6294	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
25	5, 7	nvscl 25935	. . . . . . . . . . . 12 |- ( ( U e. NrmCVec /\ k e. CC /\ A e. X ) -> ( k S A ) e. X )
26	4, 25	mp3an1 1313	. . . . . . . . . . 11 |- ( ( k e. CC /\ A e. X ) -> ( k S A ) e. X )
27	1, 26	sylan 469	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( k S A ) e. X )
28	5, 6, 7, 19, 3	ipdiri 26159	. . . . . . . . . . 11 |- ( ( ( k S A ) e. X /\ A e. X /\ B e. X ) -> ( ( ( k S A ) G A ) P B ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
29	18, 28	mp3an3 1315	. . . . . . . . . 10 |- ( ( ( k S A ) e. X /\ A e. X ) -> ( ( ( k S A ) G A ) P B ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
30	27, 29	sylancom 665	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k S A ) G A ) P B ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
31	24, 30	eqtr4d 2446	. . . . . . . 8 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) = ( ( ( k S A ) G A ) P B ) )
32	17, 31	eqtr4d 2446	. . . . . . 7 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) )
33		oveq1 6285	. . . . . . 7 |- ( ( ( k S A ) P B ) = ( k x. ( A P B ) ) -> ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
34	32, 33	sylan9eq 2463	. . . . . 6 |- ( ( ( k e. NN0 /\ A e. X ) /\ ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
35		adddir 9617	. . . . . . . . 9 |- ( ( k e. CC /\ 1 e. CC /\ ( A P B ) e. CC ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
36	2, 35	mp3an2 1314	. . . . . . . 8 |- ( ( k e. CC /\ ( A P B ) e. CC ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
37	1, 21, 36	syl2an 475	. . . . . . 7 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
38	37	adantr 463	. . . . . 6 |- ( ( ( k e. NN0 /\ A e. X ) /\ ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
39	34, 38	eqtr4d 2446	. . . . 5 |- ( ( ( k e. NN0 /\ A e. X ) /\ ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) )
40	39	exp31 602	. . . 4 |- ( k e. NN0 -> ( A e. X -> ( ( ( k S A ) P B ) = ( k x. ( A P B ) ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) ) )
41	40	a2d 26	. . 3 |- ( k e. NN0 -> ( ( A e. X -> ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( A e. X -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) ) )
42		eqid 2402	. . . . . 6 |- ( 0vec ` U ) = ( 0vec ` U )
43	5, 42, 19	dip0l 26045	. . . . 5 |- ( ( U e. NrmCVec /\ B e. X ) -> ( ( 0vec ` U ) P B ) = 0 )
44	4, 18, 43	mp2an 670	. . . 4 |- ( ( 0vec ` U ) P B ) = 0
45	5, 7, 42	nv0 25946	. . . . . 6 |- ( ( U e. NrmCVec /\ A e. X ) -> ( 0 S A ) = ( 0vec ` U ) )
46	4, 45	mpan 668	. . . . 5 |- ( A e. X -> ( 0 S A ) = ( 0vec ` U ) )
47	46	oveq1d 6293	. . . 4 |- ( A e. X -> ( ( 0 S A ) P B ) = ( ( 0vec ` U ) P B ) )
48	21	mul02d 9812	. . . 4 |- ( A e. X -> ( 0 x. ( A P B ) ) = 0 )
49	44, 47, 48	3eqtr4a 2469	. . 3 |- ( A e. X -> ( ( 0 S A ) P B ) = ( 0 x. ( A P B ) ) )
50		oveq1 6285	. . . . . 6 |- ( j = 0 -> ( j S A ) = ( 0 S A ) )
51	50	oveq1d 6293	. . . . 5 |- ( j = 0 -> ( ( j S A ) P B ) = ( ( 0 S A ) P B ) )
52		oveq1 6285	. . . . 5 |- ( j = 0 -> ( j x. ( A P B ) ) = ( 0 x. ( A P B ) ) )
53	51, 52	eqeq12d 2424	. . . 4 |- ( j = 0 -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( 0 S A ) P B ) = ( 0 x. ( A P B ) ) ) )
54	53	imbi2d 314	. . 3 |- ( j = 0 -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( 0 S A ) P B ) = ( 0 x. ( A P B ) ) ) ) )
55		oveq1 6285	. . . . . 6 |- ( j = k -> ( j S A ) = ( k S A ) )
56	55	oveq1d 6293	. . . . 5 |- ( j = k -> ( ( j S A ) P B ) = ( ( k S A ) P B ) )
57		oveq1 6285	. . . . 5 |- ( j = k -> ( j x. ( A P B ) ) = ( k x. ( A P B ) ) )
58	56, 57	eqeq12d 2424	. . . 4 |- ( j = k -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) )
59	58	imbi2d 314	. . 3 |- ( j = k -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) ) )
60		oveq1 6285	. . . . . 6 |- ( j = ( k + 1 ) -> ( j S A ) = ( ( k + 1 ) S A ) )
61	60	oveq1d 6293	. . . . 5 |- ( j = ( k + 1 ) -> ( ( j S A ) P B ) = ( ( ( k + 1 ) S A ) P B ) )
62		oveq1 6285	. . . . 5 |- ( j = ( k + 1 ) -> ( j x. ( A P B ) ) = ( ( k + 1 ) x. ( A P B ) ) )
63	61, 62	eqeq12d 2424	. . . 4 |- ( j = ( k + 1 ) -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) )
64	63	imbi2d 314	. . 3 |- ( j = ( k + 1 ) -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) ) )
65		oveq1 6285	. . . . . 6 |- ( j = N -> ( j S A ) = ( N S A ) )
66	65	oveq1d 6293	. . . . 5 |- ( j = N -> ( ( j S A ) P B ) = ( ( N S A ) P B ) )
67		oveq1 6285	. . . . 5 |- ( j = N -> ( j x. ( A P B ) ) = ( N x. ( A P B ) ) )
68	66, 67	eqeq12d 2424	. . . 4 |- ( j = N -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( N S A ) P B ) = ( N x. ( A P B ) ) ) )
69	68	imbi2d 314	. . 3 |- ( j = N -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) ) ) )
70	41, 49, 54, 59, 64, 69	nn0indALT 10999	. 2 |- ( N e. NN0 -> ( A e. X -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) ) )
71	70	imp 427	1 |- ( ( N e. NN0 /\ A e. X ) -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) )

Syntax hints:   -> wi 4   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   ` cfv 5569  (class class class)co 6278   CCcc 9520   0cc0 9522   1c1 9523   + caddc 9525   x. cmul 9527   NN0cn0 10836   NrmCVeccnv 25891   +vcpv 25892   BaseSetcba 25893   .sOLDcns 25894   0veccn0v 25895   .iOLDcdip 26024   CPreHil OLDccphlo 26141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-sum 13658  df-grpo 25607  df-gid 25608  df-ginv 25609  df-ablo 25698  df-vc 25853  df-nv 25899  df-va 25902  df-ba 25903  df-sm 25904  df-0v 25905  df-nmcv 25907  df-dip 26025  df-ph 26142

This theorem is referenced by:  ipasslem2  26161  ipasslem3  26162  ipasslem4  26163